Continuous function spaces, (db)-spaces and strongly Hewitt spaces (Q1878519)

The authors consider here some peculiar properties of Tychonoff spaces \(X\) and corresponding real function spaces \(C(X)\), exhibiting connections between them. The problems discussed concern the so-called property (db): if \(E\) is the union of an increasing sequence \((E_n)\) of vector spaces, then some \(E_n\) is dense and barreled. The authors solve here the long standing problem of characterizing those Tychonoff spaces \(X\) for which the function space \(C_k(X)\) with the compact-open topology satisfies property (db). The authors consider also some results related to strongly Hewitt spaces, give three examples of spaces with special properties and some questions and problems.